Probably fix what's wrong.
No
Not if they test emissions.
Get it fixed.
No, the odometer is not part of an emissions test.
Replace the catalytic converter.
If it didn't have emissions controls from the factory you don't have to pass any emissions test, check with your state as some are different, especially California.
After one year, all new cars bought must pass the emissions test in counties that require the test. Once a vehicle is 25 years old, the emissions test is no longer required.
No, it will not.
put a cat on it
If it is leaking or rotted
Probably not as the emissions test (Smog test) is carried out by inserting a sensor into the exhaust pipe.
About 60 miles in order for the computer to log emissions information to relay to the testing computer.